"would" usually implies a conditional, even if it's not stated explicitly.

The standard versions of these four are:

1. That would be cool if you could do that.
2. It would be nice of you if you could do that.
3. That will be cool if you can do that.
4. It will be nice of you if you can do that.

1 and 2 are more polite, but 3 and 4 are not by any means impolite.
1 and 2 indicate that the speaker is less certain that he will get what he wants.
3 and 4 indicate that the speaker is more confident that he will get what he wants.

In 1 and 2, the speaker is trying to get something that he wants.
In 3 and 4, the same thing happens, but it gives the impression that someone has already offered something to the speaker, and the speaker is confirming that it will be done, and, in a way, giving his thanks for the offer.

Your original 1 and 2 are "mixed" stuctures: "would" mixed with "can".
These are possible in casual conversations. Some people do speak that way. However, I recommend learning to recognize it when you hear it, but not "mixing" when you speak or write -- at least not until you gain more experience with these structures.

I hope that makes sense.

It's not always clear what is grammatical and what is ungrammatical. Sometimes it depends on which author you read. In your place I would not use this kind of combination of tenses. I believe most authors would say it's possible but not standard. This means they consider it grammatical, but not the best way to express it.

1 uses "could" to mean "may" or "might", i.e., "It might be anyone" = "It is possible that it is anyone". I would not consider this part of a conditional structure, no.
(However, the form "could" can be thought of as a conditional form when used as such, as in "If I were taller, I could (would be able to) touch the ceiling".)

2 is more like volitional "would" ("Why would I want to lie?"), but also has aspects of conditional "would" ("Under what conditions would I lie?", i.e., *I would lie IF what?").

3 and 4 mean the same as they do with "could" instead of "can".

5 means "That would be cool if you were able to do that."
6 is another 'mixed' sentence, and is a rather poor way to phrase 5. I suppose both 5 and 6 would mean the same thing. 6 isn't used enough to have a meaning separate from 5.

In "if" sentences like this one, we have the three standard "conditional structures". I don't think you should use other combinations, especially on tests!

(Real condition)
It will be nice if you do that.
It will be nice if you can do that.
(Unreal condition)
It would be nice if you did that.
It would be nice if you could do that.
(Counterfactual)
It would have been nice if you had done that. (You didn't do it.)
It would have been nice if you could have done that. (You were not able to do it.)

CJ

I was trying to show you the most typical uses. You can probably find all possible combinations somewhere. Without writing an entire book or two on the subject, it would be impossible to discuss them all.

When reporting speech, you can use the present introduced by the past if what you are reporting is "timelessly" true. You can, but you don't have to.

If it is simply a fact that such-and-such is illegal, you can say "He said that it is illegal".
But you'll be correct with the standard tense matching as well: "He said that it was illegal".

Note that "He is going to the party" is not a "timeless" or "eternal" truth, but if said quite soon after he said, "I can go to the party" or if said before the party has even taken place yet, you can say "He said he can go to the party". But you don't have to. The standard is perfectly OK and means the same thing: "He said he could go to the party."

"needing help" can be thought of as "timeless" - a fact - and it also implies that something should happen ("getting help") in the future, so if help has not yet been obtained, you can say "He said you need some help", but the standard form is fine: "He said you needed some help".

Your first question "Why would she ..." is correct as is. It's a question, and I suspect that being a question may make a difference. And no, I don't know exactly what the differences are!

You only use quotation marks when you repeat word for word what was said.

If you actually said the words "I forgot", I can write either of the following:

You said, "I forgot". (You actually said the word "I" when you spoke.)
You said that you forgot. (You did not use the word "you" when you spoke.)
You said you forgot. (Same as previous but without "that".)

In general, try to realize that you can't necessarily learn the modals very well by relying on a few vague rules which have lots of exceptions - which is about the best anyone can help you with when it comes right down to it. Only by reading English and conversing with English speakers over the months and years will it ever make sense to you. "Modals" is a complicated subject and probably best learned by imitation rather than rules - or at least with a combination of the two.